As shown in FIG. 1, a typical bandgap reference voltage generator has a self-bias circuit 10 and a start-up circuit 12 for starting up the bandgap reference voltage generator. In the self-bias circuit 10, two MOSFETs M1 and M2 have control terminals connected to each other and to an output terminal VC of an operational amplifier 14, a resistor R1 and a bipolar junction transistor (BJT) Q1 that is configured as a diode are serially connected between a positive input terminal VA of the operational amplifier 14 and a ground terminal GND, a BJT Q2 that is configured as a diode is connected between a negative input terminal VB of the operational amplifier 14 and the ground terminal GND, a resistor R2 and the MOSFET M1 are serially connected between a power supply terminal VDD and the positive input terminal VA of the operational amplifier 14, and a resistor R3 and the MOSFET M2 are serially connected between the power supply terminal VDD and the negative input terminal VB of the operational amplifier 14. The resistors R2 and R3 have equal resistances. In the start-up circuit 12, a MOSFET M3 is connected between the power supply terminal VDD and the negative input terminal VB of the operational amplifier 14, a MOSFET M4 is connected between the power supply terminal VDD and a control terminal VD of the MOSFET M3 in association with the MOSFET M1 to establish a current mirror, and a MOSFET M5 is connected between the control terminal VD of the MOSFET M3 and the ground terminal GND and has a control terminal connected to the power supply terminal VDD.
When a supply voltage VDD is applied to start the bandgap reference voltage generator of FIG. 1, the MOSFETs M3 and M5 are turned on, the MOSFET M5 is equivalent to a resistor, and the negative input terminal VB of the operational amplifier 14 is connected to the power supply terminal VDD through the MOSFET M3 so that the voltage VB on the negative input terminal is pulled high, causing the output voltage VC of the operational amplifier 14 decreasing and thereby turning on the MOSFET M1 to generate a current I1 that will be mirrored by the MOSFET M4 to generate a current I3 and thus pull high the control terminal voltage VD of the MOSFET M3. Once the voltage VD increases to a certain threshold, the MOSFET M3 is turned off and thus turns off the start-up circuit 12, thereby finishing the start-up process of the bandgap reference voltage generator.
When the bandgap reference voltage generator of FIG. 1 is at steady state, the operational amplifier 14 maintains the voltages of the two input terminals thereof asVA=VB=Vbe,  [Eq-1]where Vbe is the emitter-base voltage of the BJT Q2, which has a negative temperature coefficient. The MOSFETs M1 and M2 have equal size, and the BJTs Q1 and Q2 has a size ratio of N:1, so thatI2=I1=[VT×ln(N)]/R1,  [Eq-2]where VT is the thermal voltage, which has a positive temperature coefficient. Since the resistors R2 and R3 have equal resistances, the output terminal 16 of the bandgap reference voltage generator provides an output voltage
                                                        Vbg              =                            ⁢                                                I                  ⁢                                                                          ⁢                  2                  ×                  R                  ⁢                                                                          ⁢                  3                                +                Vbe                                                                                                        =                                ⁢                                                      [                                          VT                      ×                                              ln                        ⁡                                                  (                          N                          )                                                                    ×                      R                      ⁢                                                                                          ⁢                                              2                        /                        R                                            ⁢                                                                                          ⁢                      1                                        ]                                    +                  Vbe                                            ,                                                          [                  Eq          ⁢                      -                    ⁢          3                ]            which hints that the temperature coefficient of the voltage Vbg can be zero by adjusting the ratio R2/R1. However, due to Vbe, only when the output voltage Vbg is approximately 1.24 V, can the temperature coefficient be zero, so that the bandgap reference voltage generator of FIG. 1 cannot work under a low power supply voltage, for example, VDD=1V.
FIG. 2 is a conventional low-voltage bandgap reference voltage generator, in which the self-bias circuit 10 of FIG. 1 is modified by moving the resistors R2 and R3 to be respectively connected between the positive input terminal VA of the operational amplifier 14 and the ground terminal GND and between the negative input terminal VB of the operational amplifier 14 and the ground terminal GND, adding a MOSFET M6 and a resistor R4 serially connected between the power supply terminal VDD and the ground terminal GND, and establishing a current mirror by the MOSFETs M1 and M6.
When a supply voltage VDD is applied to start up the bandgap reference voltage generator of FIG. 2, the MOSFETs M3 and M5 are turned on, the negative input terminal voltage VB of the operational amplifier 14 is pulled high, the output voltage VC of the operational amplifier 14 decreases, the MOSFET M1 is turned on to generate a current I5, the MOSFET M4 mirrors the current I5 to generate a current I3, and the voltage VD increases. Once the voltage VD increases to a certain threshold, the MOSFET M3 is turned off and thus turns off the start-up circuit 12, thereby finishing the start-up process of the bandgap reference voltage generator.
When the bandgap reference voltage generator of FIG. 2 is at steady state, the operational amplifier 14 maintains its input voltages VA and VB to be equal to each other as shown in the equation Eq-1, so that the current I4 of the resistor R2 is equal to Vbe/R2, and due to the resistors R2 and R3 having equal resistances, the currents I1 and 12 are equal to each other as shown in the equation Eq-2, resulting inI5=I1+I4=VT×ln(N)/R1+Vbe/R2.  [Eq-4]If the MOSFETs M1 and M6 have equal size, the output terminal 18 of the bandgap reference voltage generator will provide the output voltage
                                                        Vbg              =                            ⁢                                                I                  ⁢                                                                          ⁢                  6                  ×                  R                  ⁢                                                                          ⁢                  4                                =                                  I                  ⁢                                                                          ⁢                  5                  ×                  R                  ⁢                                                                          ⁢                  4                                                                                                                        =                                ⁢                                                      (                                          R                      ⁢                                                                                          ⁢                                              4                        /                        R                                            ⁢                                                                                          ⁢                      2                                        )                                    ×                                      [                                          Vbe                      +                                              VT                        ×                                                  ln                          ⁡                                                      (                            N                            )                                                                          ×                                                  (                                                      R                            ⁢                                                                                                                  ⁢                                                          2                              /                              R                                                        ⁢                                                                                                                  ⁢                            1                                                    )                                                                                      ]                                                              ,                                                          [                  Eq          ⁢                      -                    ⁢          5                ]            which hints that the output voltage Vbg can be independent of temperature by adjusting the ratio R2/R1, and can be adjusted with its level by adjusting the ratio R4/R2. As illustrated in FIG. 1, to neutralize the negative temperature coefficient of the voltage Vbe and the positive temperature coefficient of the thermal voltage VT, the term Vbe+VT×ln(N)×(R2/R1) in the equation Eq-5 must be approximately 1.24 V. If it is set (R4/R2)=⅔, the output voltage Vbg is approximately 0.8V, and thus the bandgap reference voltage generator of FIG. 2 may still work normally even VDD=1V. However, when the bandgap reference voltage generator of FIG. 2 is just started up, the BJTs Q1 and Q2 that are configured as diodes are not turned on yet, and thus the current of the self-bias circuit totally flows through the resistors R2 and R3. If the start-up circuit 12 is turned off when the BJTs Q1 and Q2 are still off, an incorrect output voltage Vbg will be generated. Therefore, the start-up circuit 12 and the on time of the MOSFET M3 must be carefully designed to enable the bandgap reference voltage generator to be correctly started up, which, however, prolongs the start-up time.
In addition, the self-bias circuit of FIG. 1 needs two MOSFETs M1 and M2 to establish the current I1, and the self-bias circuit of FIG. 2 also needs two MOSFETs M1 and M2 to establish the current I5, so that error may be occurred if the MOSFETs M1 and M2 are not matched to each other.
U.S. Pat. No. 6,906,581 provides a bandgap reference voltage generator that includes two current generators for respectively providing a first current having a positive temperature coefficient and a second current having a negative temperature coefficient, and an output resistor for generating an output voltage independent of temperature according to the first and second currents. Although this bandgap reference voltage generator may work when the supply voltage is lower than 1.24 V, and may be started up fast, the self-bias circuit thereof still needs two MOSFETs to establish the first current having the positive temperature coefficient, so that error still may be occurred if the two MOSFETs are not matched to each other.